12.3 Problems on variance, covariance, linear regression

(See Exercise 1 from "Problems on Distribution and Density Functions ", and Exercise 1 from "Problems on Mathematical Expectation", m-file npr07_01.m ). The class $\{C_j:1\le j\le 10\}$ is a partition. Random variable X has values $\{1,3,2,3,4,2,1,3,5,2\}$ on C ₁ through C ₁₀ , respectively, with probabilities 0.08, 0.13, 0.06, 0.09, 0.14, 0.11, 0.12, 0.07, 0.11, 0.09.Determine $\mathrm{Var}\left[X\right]$ .

(See Exercise 2 from "Problems on Distribution and Density Functions ", and Exercise 2 from "Problems on Mathematical Expectation", m-file npr07_02.m ). A store has eight items for sale. The pricesare $3.50, $5.00, $3.50, $7.50, $5.00, $5.00, $3.50, and $7.50, respectively. A customer comes in. She purchasesone of the items with probabilities 0.10, 0.15, 0.15, 0.20, 0.10 0.05, 0.10 0.15. The random variable expressing the amount of her purchase may be written

Determine $\mathrm{Var}\left[X\right]$ .

(See Exercise 12 from "Problems on Random Variables and Probabilities", Exercise 3 from "Problems on Mathematical Expectation", m-file npr06_12.m ). The class $\{A,B,C,D\}$ has minterm probabilities

Consider $X=I_A+I_B+I_C+I_D$ , which counts the number of these events which occur on a trial. Determine $\mathrm{Var}\left[X\right]$ .

(See Exercise 4 from "Problems on Mathematical Expectation"). In a thunderstorm in a national park there are 127 lightning strikes. Experience shows that the probability of each lightning strike starting a fire is about0.0083. Determine $\mathrm{Var}\left[X\right]$ .

(See Exercise 5 from "Problems on Mathematical Expectation"). Two coins are flipped twenty times. Let X be the number of matches (both heads or both tails). Determine $\mathrm{Var}\left[X\right]$ .

(See Exercise 6 from "Problems on Mathematical Expectation"). A residential College plans to raise money by selling “chances” on a board. Fifty chances are sold. A player pays $10 to play; he or she wins $30with probability $p=0.2$ . The profit to the College is

Determine $\mathrm{Var}\left[X\right]$ .

(See Exercise 7 from "Problems on Mathematical Expectation"). The number of noise pulses arriving on a power circuit in an hour is a random quantity X having Poisson (7) distribution. Determine $\mathrm{Var}\left[X\right]$ .

(See Exercise 24 from "Problems on Distribution and Density Functions", and Exercise 8 from "Problems on Mathematical Expectation"). The total operating time for the units in Exercise 24 from "Problems on Distribution and Density Functions" is a random variable $T\sim$ gamma (20, 0.0002). Determine $\mathrm{Var}\left[T\right]$ .